In the chapter, Approximation Algorithms of Introduction to Algorithm, 3rd Edition, for the approximation problem Travelling Salesman Problem, the author proposes a approximation method that first constructs a minimum spanning tree.
In order to prove this algorithm is a 2-approximation algorithm, the author claims that:
The weight of the minimum spanning tree $T$ is less than the cost of the optimal tour.
I am wondering if the minimum spanning tree(which is acyclic) of $G$ ensures that its weight is necessarily smaller than any tour(which is cyclic) of the same graph $G$
The original claim is: